ALGORITHM FOR THE EVALUATION OF THE COMPLEX AIRY FUNCTIONS

The evaluation of complex Airy functions is required in the approximation of certain second-order linear differential equations arising in the treatment of multiple turning-point and energy curve-crossing problems in quantum mechanics. Pairs of numerically linearly independent solutions throughout the z-plane can be constructed from the fundamental solutions to the complex Airy equation, Ai(z), Bi(z), and Ai(z e±2πi/3). Integral representations for these complex functions and their derivatives are given, and being of the Stieltjes type, the integrals are evaluated using the generalized Gaussian quadrature method of Shohat and Tamarkin as implemented by Gordon. These integral representations, employed together with the Taylor series for small z and the appropriate connection formulas, allow the creation of an accurate and efficient algorithm to evaluate the complex functions over the entire z-plane. The algorithm is presented in detail at the end of this article. © 1979.